A diverging lens with a focal length of -15 cm and a converging lens with a focal length of 12

step 1 This problem covers the concept of the lens formula which says that 1 upon O plus 1 upon I equal

A diverging lens with a focal length of -15 cm and a...

A diverging lens with a focal length of $-15 \mathrm{~cm}$ and a converging lens with a focal length of $12 \mathrm{~cm}$ have a common central axis. Their separation is $12 \mathrm{~cm}$. An object of height $1.0 \mathrm{~cm}$ is $10 \mathrm{~cm}$ in front of the diverging lens, on the common central axis. (a) Where does the lens combination produce the final image of the object (the one produced by the second, converging lens)? (b) What is the height of that image? (c) Is the image real or virtual? (d) Does the image have the same orientation as the object or is it inverted?

Key Concepts

Sign Conventions for Lenses

Sign conventions are critical when using the thin lens equation. They determine the positive and negative directions for object distances, image distances, and focal lengths. For instance, converging lenses typically have positive focal lengths while diverging lenses have negative ones. These conventions also help distinguish between real and virtual images created by lenses.

Real vs. Virtual Images

The concepts of real and virtual images in optics are determined by the sign of the image distance. A real image is formed when light rays actually converge and can be projected onto a screen, typically yielding a positive image distance according to the used sign convention, whereas a virtual image is formed when rays appear to diverge from a point behind the lens, indicated by a negative image distance.

Magnification and Image Orientation

Magnification in lens systems, defined as the ratio of the image height to the object height (and sometimes given by m = -di/do for single lenses), provides information about the size and orientation of the image. A negative magnification indicates an inverted image, while the overall magnification in multi-lens systems is the product of the individual magnifications.

Thin Lens Equation

The thin lens equation, expressed as 1/f = 1/do + 1/di, relates the focal length of a lens (f) to the object distance (do) and the image distance (di). This equation is fundamental for determining the position of the image produced by a single lens and is the basis for understanding more complex lens systems.

Sequential Analysis in Multi-lens Systems

When multiple lenses are present, the image formed by one lens becomes the object for the next lens in the system. This sequential analysis requires careful consideration of the distances between lenses and may involve adjusting the object distance for the subsequent lens based on the location of the first image relative to that lens.

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